Wavelet and Wave Analysis As Applied to Materials With Micro or Nanostructure

by Cattani, Carlo; Rushchitsky, Jeremiah

• ISBN13: 9789812707840
• ISBN10: 9812707840
• Format: Hardcover
• Copyright: 2007-09-06
• Publisher: World Scientific Pub Co Inc

Table of Contents

Preface	p. V
Introduction	p. 1
Wavelet Analysis	p. 13
Wavelet and Wavelet Analysis. Preliminary Notion	p. 13
The space L[superscript 2](R)	p. 15
The spaces L[superscript p](R)(p[greater than or equal]1)	p. 16
The Hardy spaces H[superscript p](R)(p[greater than or equal]1)	p. 17
The sketch scheme of wavelet analysis	p. 18
Rademacher, Walsh and Haar Functions	p. 26
System of Rademacher functions	p. 26
System of Walsh functions	p. 28
System of Haar functions	p. 32
Integral Fourier Transform. Heisenberg Uncertainty Principle	p. 44
Window Transform. Resolution	p. 52
Examples of window functions	p. 54
Properties of the window Fourier transform	p. 57
Discretization and discrete window Fourier transform	p. 59
Bases. Orthogonal Bases. Biorthogonal Bases	p. 63
Frames. Conditional and Unconditional Bases	p. 71
Wojtaszczyk's definition of unconditional basis (1997)	p. 81
Meyer's definition of unconditional basis (1997)	p. 82
Donoho's definition of unconditional basis (1993)	p. 82
Definition of conditional basis	p. 82
Multiresolution Analysis	p. 83
Decomposition of the Space L[superscript 2](R)	p. 95
Discrete Wavelet Transform. Analysis and Synthesis	p. 109
Analysis: transition from the fine scale to the coarse scale	p. 111
Synthesis: transition from the coarse scale to the fine scale	p. 113
Wavelet Families	p. 116
Haar wavelet	p. 117
Stromberg wavelet	p. 120
Gabor wavelet	p. 123
Daubechies-Jaffard-Journe wavelet	p. 123
Gabor-Malvar wavelet	p. 124
Daubechies wavelet	p. 125
Grossmann-Morlet wavelet	p. 126
Mexican hat wavelet	p. 127
Coifman wavelet - coiflet	p. 128
Malvar-Meyer-Coifman wavelet	p. 130
Shannon wavelet or sinc-wavelet	p. 130
Cohen-Daubechies-Feauveau wavelet	p. 131
Geronimo-Hardin-Massopust wavelet	p. 132
Battle-Lemarie wavelet	p. 133
Integral Wavelet Transform	p. 137
Definition of the wavelet transform	p. 137
Fourier transform of the wavelet	p. 138
The property of resolution	p. 139
Complex-value wavelets and their properties	p. 141
The main properties of wavelet transform	p. 141
Discretization of the wavelet transform	p. 142
Orthogonal wavelets	p. 143
Dyadic wavelets and dyadic wavelet transform	p. 144
Equation of the function (signal) energy balance	p. 144
Materials with Micro- or Nanostructure	p. 147
Macro-, Meso-, Micro-, and Nanomechanics	p. 147
Main Physical Properties of Materials	p. 156
Thermodynamical Theory of Material Continua	p. 160
Composite Materials	p. 168
Classical Model of Macroscopic (Effective) Moduli	p. 174
Other Microstructural Models	p. 181
Bolotin model of energy continualization	p. 182
Achenbach-Hermann model of effective stiffness	p. 183
Models of effective stiffness of high orders	p. 184
Asymptotic models of high orders	p. 185
Drumheller-Bedford lattice microstructural models	p. 186
Mindlin microstructural theory	p. 187
Eringen microstructural model. Eringen-Maugin model	p. 188
Pobedrya microstructural theory	p. 190
Structural Model of Elastic Mixtures	p. 191
Viscoelastic mixtures	p. 210
Piezoelastic mixtures	p. 213
Computer Modelling Data on Micro- and Nanocomposites	p. 216
Waves in Materials	p. 229
Waves Around the World	p. 229
Analysis of Waves in Linearly Deformed Elastic Materials	p. 232
Volume and shear elastic waves in the classical approach	p. 232
Plane elastic harmonic waves in the classical approach	p. 237
Cylindrical elastic waves in the classical approach	p. 241
Volume and shear elastic waves in the nonclassical approach	p. 244
Plane elastic harmonic waves in the nonclassical approach	p. 247
Analysis of Waves in Nonlinearly Deformed Elastic Materials	p. 253
Basic notions of the nonlinear theory of elasticity. Strains	p. 253
Forces and stresses	p. 260
Balance equations	p. 262
Nonlinear elastic isotropic materials. Elastic Potentials	p. 267
Nonlinear Wave Equations	p. 276
Nonlinear wave equations for plane waves. Methods of solving	p. 276
Method of successive approximations	p. 281
Method of slowly varying amplitudes	p. 283
Nonlinear wave equations for cylindrical waves	p. 285
Comparison of Murnaghan and Signorini Approaches	p. 308
Comparison of some results for plane waves	p. 308
Comparison of cylindrical and plane wave in the Murnaghan model	p. 322
Simple and Solitary Waves in Materials	p. 337
Simple (Riemann) Waves	p. 337
Simple waves in nonlinear acoustics	p. 337
Simple waves in fluids	p. 340
Simple waves in the general theory of waves	p. 344
Simple waves in mechanics of electromagnetic continua	p. 345
Solitary Elastic Waves in Composite Materials	p. 346
Simple solitary waves in materials	p. 346
Chebyshev-Hermite functions	p. 347
Whittaker functions	p. 349
Mathieu functions	p. 352
Interaction of simple waves. Self-generation	p. 353
The solitary wave analysis	p. 359
New Hierarchy of Elastic Waves in Materials	p. 373
Classical harmonic waves (periodic, nondispersive)	p. 374
Classical arbitrary elastic waves (D'Alembert waves)	p. 374
Classical harmonic elastic waves (periodic, dispersive)	p. 375
Nonperiodic elastic solitary waves (with the phase velocity depending on the phase)	p. 377
Simple elastic waves (with the phase velocity depending on the amplitude)	p. 379
Solitary Waves and Elastic Wavelets	p. 381
Elastic Wavelets	p. 381
The Link between the Trough Length and the Characteristic Length	p. 391
Initial Profiles as Chebyshev-Hermite and Whittaker Functions	p. 396
Some Features of the Elastic Wavelets	p. 410
Solitary Waves in Mechanical Experiments	p. 422
Ability of Wavelets in Detecting the Profile Features	p. 435
Bibliography	p. 443
Index	p. 455
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